Embedding finitely generated Abelian lattice-ordered groups: Higman's Theorem and a realisation of \pi
by
A.M.W. Glass
Cambridge, UK
Coauthors: Vincenzo Marra

Graham Higman proved that a finitely generated group can be embedded in a finitely presented group iff it has a recursively enumerable set of defining relations. We consider the analogue for lattice-ordered groups. Clearly, the finitely generated lattice-ordered groups that can be l-embedded in finitely presented lattice-ordered groups must have recursively enumerable sets of defining relations. We prove the converse direction for a special class of lattice-ordered groups:

Theorem. Every finitely generated Abelian lattice-ordered group that has finite rank and a recursively enumerable set of defining relations can be \l-embedded in a finitely presented lattice-ordered group.

As a consequence we obtain that D(\pi) the Abelian rank 2 group \Z² with order (m, n) > 0   iff  m+n\pi > 0 can be l-embedded in a finitely presented lattice-ordered group, whence \pi is ``l-algebraic'' in that it can be captured by finitely many relations in this language. Indeed,

Corollary. The recursive reals are precisely those real numbers \xi for which D(\xi) can be \l-embedded in a finitely presented lattice-ordered group.

The technique is an amalgamation of three disparate areas: (1) continued fractions, (2) recent advances in direct limits of Abelian lattice-ordered groups, and (3) using permutation groups to encode the necessary information (a technique whose origins can be found in work of Ralph McKenzie and Richard Thompson).

Date received: January 9, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caig-59.